where the first equality follows from [1]}, the second equality follows from [1]} and the third equality follows from [1]}. We also consider a constant system density \(\eta =\mathcal {O}\left(\omega \right)\)  following the setting of [1]}. Applying [thm:fermionicseminorm]Theorem thm:fermionicseminorm, we find that a \(p\) th-order formula \({S}_p(t)\)  can approximate the evolution of electronic-structure Hamiltonian with Trotter error
\(\begin{aligned}\left\Vert {S}_p(t)-e^{-itH}\right\Vert _\eta &=\mathcal {O}\left(\left(\left\Vert \tau \right\Vert +\left\Vert \nu \right\Vert _{\max }\eta \right)^{p-1}\left\Vert \tau \right\Vert \left\Vert \nu \right\Vert _{\max }\eta ^2 t^{p+1}\right)\\&=\mathcal {O}\left(\left(\frac{n^{2/3}}{\eta ^{2/3}}+n^{1/3}\eta ^{2/3}\right)^{p}n^{1/3}\eta ^{2/3} t^{p+1}\right).\end{aligned}\) 
